Math 302 Modern Algebra Spring 2009 Homework Assignments Assignment 7 due Monday, April 6 4.3 Irreducibles and Unique Factorization 6. Show that x 2 + 1 is irreducible in Q [ x ]. [ Hint: Use a proof by contradiction. Suppose x 2 + 1 = ( ax + b )( cx + d ) for some a, b, c, d ∈ Q . Show this leads to a contradiction.] 9. Find all irreducible polynomials of (b) degree 3 in Z 2 [ x ]. Be sure to indicate why each of these is in fact irreducible. (c) degree 2 in Z 3 [ x ]. Be sure to indicate why each of these is in fact irreducible. 16. Let F be a ±eld. Prove: For all p ( x ) ∈ F [ x ], p ( x ) is irreducible in F [ x ] if and only if for every g ( x ) ∈ F [ x ], either p ( x ) | g ( x ) or p ( x ) and g ( x ) are relatively prime. 22. (a) Show that x 3 + a is reducible in Z 3 [ x ] for each a ∈ Z 3 . 4.4 Polyomial Functions, Roots, and Reducibility 3. Use the Remainder Theorem to determine if h ( x ) is a factor of f ( x ). (b)
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